Abstract
In this paper, we study a convergence theorem for a finite second-order Markov chain indexed by a general infinite tree with uniformly bounded degree. Meanwhile, the strong law of large numbers (LLN) and Shannon-McMillan theorem for a finite second-order Markov chain indexed by this tree are obtained. 2000 Mathematics Subject Classification: 60F15; 60J10.
Highlights
A tree is a graph G = {T, E} that is connected and contains no circuits
Takacs has studied the strong law of large numbers for the univariate functions of finite Markov chains indexed by an infinite tree with uniformly bounded degree
Huang and Yang have studied the strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree, which generalize the result of [8]
Summary
Takacs (see [8]) has studied the strong law of large numbers for the univariate functions of finite Markov chains indexed by an infinite tree with uniformly bounded degree. Huang and Yang (see [10]) have studied the strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree, which generalize the result of [8]. We first study a convergence theorem for a finite second-order Markov chain indexed by a general infinite tree with uniformly bounded degree. We obtain the strong law of large numbers and ShannonMcMillan theorem for a class of finite second-order Markov chain indexed by a general infinite tree with uniformly bounded degree.
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